Efficient Ranking-Based Whale Optimizer for Parameter Extraction of Three-Diode Photovoltaic Model: Analysis and Validations
Abstract
:1. Introduction
- Slow convergence toward the best solution because of the distance control factor that reduces gradually with the iterations;
- Using the current positions in the next generation, even if it is worse, may reduce the probability of getting to better solutions;
- After half the maximum iterations, the exploration operator will be terminated and hence stagnation inside local minima is inevitable if the best-so-far solution is so.
- Proposing a new updating scheme to replace the unbeneficial solutions under the ranking method for reducing the probability of stagnation into local optimal and then integrating the WOA with this strategy in a variant named RWOA to utilize each solution within the optimization process as possible;
- Developing a novel strategy called a cyclic exploration and exploitation strategy to promote both local and global search of the RWOA for reaching further better outcomes in a new variant of RWOA called HWOA;
- Our findings show that RWOA is competitive in some cases and superior in the others in terms of final accuracy and convergence speed compared to five well-established algorithms for estimating the unknown parameters of RTC France and two PV modules (Photowatt-PWP201 and Kyocera KC200GT); however, the HWOA can be superior in all cases.
2. Mathematical Formulation of the Three-Diode Model
3. Proposed Approaches: RWOA and HWOA
WOA, Overview
- The first strategy is based on using a spiral shape to spin around their prey;
- The second uses a shrinking circle—called an encircling mechanism—to attack the prey.
- Low premature convergence toward the best solution;
- Local minima problem.
- Each whale in the population is used as much as possible based on the ranking method suggested in [20];
- The whales selected using the previous method are replaced by a novel formula to accelerate convergence;
- Memory saving is used to avoid reducing the diversity between the individuals of the population and subsequently reducing the probability of becoming trapped into local minima;
- Finally, to distribute the whales efficiently within the boundary of the optimization problem, ten chaotic maps are investigated for their suitability to be incorporated in the proposed approach.
Algorithm 1 Initialization (N, d, LB, UB) |
1. create an array W of size $NXd$ 2. //initialization 3. for i = 1: N 4. for j = 1: d 5. create a random number r between 0 and 1; 6. W (i + 1, j) = LB(j) + r × (UB(j) − LB(j)); 7. end for 8. end for 9. Return W. |
Algorithm 2 The Steps of RWOA |
1. Calling initialization (N, d, LB, UB) 2. $\overrightarrow{OW}={\overrightarrow{W}}_{i}$. 3. $\overrightarrow{f}:$ Evaluate the fitness of $W$ using Equation (17). 4. $\overrightarrow{of}=\overrightarrow{f}$, set the current fitness values in a vector called old fitness, $\overrightarrow{of}$. 5. $\overrightarrow{R}$: is a ranking vector of N cells with an initial value 0 6. Find the best whale $\overrightarrow{{W}^{\ast}}$ 7. $t=1$ 8. while ($t$ < ${t}_{max}$) 9. for each $i$ whale 10. Update a, A, p, C, and l 11. if ($p<0.5$) 12. if ($\left|A\right|<1$) 13. Update $\overrightarrow{{W}_{i}}\left(t+1\right)$ using Equation (4) 14. else 15. Update $\overrightarrow{{W}_{i}}\left(t+1\right)$ using Equation (11) 16. end if 17. else 18. Update $\overrightarrow{{W}_{i}}\left(t+1\right)$ using Equation (9) 19. end if 20. If ${R}_{i}NI$ 21. Calculate $\overrightarrow{CF}$ using Equation (14) 22. Update $\overrightarrow{{W}_{i}}$ using Equation (16) 23. End 24. If ${f}_{i}$<$of$ 25. $o{f}_{i}={f}_{i}$ 26. $\overrightarrow{O{W}_{i}}=\overrightarrow{{W}_{i}}$ 27. ${R}_{i}=0$ 28. Else 29. $\overrightarrow{{W}_{i}}=\overrightarrow{O{W}_{i}}$ 30. ${R}_{i}++$ 31. End 32. ${f}_{i}:$ Calculating the fitness of the $\overrightarrow{{W}_{i}}$ using Equation (17) 33. end for 34. Update the best whale $\overrightarrow{{W}^{\ast}}$ with $\overrightarrow{W}\left(t+1\right)$ if better 35. $t$++ 36. end while. |
Algorithm 3 The Steps of HWOA |
1. Calling initialization (N, d, LB, UB) 2. $\overrightarrow{OW}={\overrightarrow{W}}_{i}$. 3. $\overrightarrow{f}:$ Evaluate the fitness of $W$ using Equation (17) 4. $\overrightarrow{of}=\overrightarrow{f}$, set the current fitness values in a vector called old fitness, $\overrightarrow{of}$ 5. $\overrightarrow{R}$: is a ranking vector of N cells with an initial value 0 6. Find the best whale $\overrightarrow{{W}^{\ast}}$ 7. $t=1$ 8. while ($t$ < ${t}_{max}$) 9. for each $i$ whale 10. Update a, A, p, C, and l 11. if ($p<0.5$) 12. if ($\left|A\right|<1$) 13. Update $\overrightarrow{{W}_{i}}\left(t+1\right)$ using Equation (4) 14. else 15. Update $\overrightarrow{{W}_{i}}\left(t+1\right)$ using Equation (11) 16. end if 17. else 18. Update $\overrightarrow{{W}_{i}}\left(t+1\right)$ using Equation (9) 19. end if 20. If ${R}_{i}NI$ 21. Calculate $\overrightarrow{CF}$ using Equation (14) 22. Update $\overrightarrow{{W}_{i}}$ using Equation (16) 23. End 24. ${f}_{i}:$ Calculating the fitness of the $\overrightarrow{{W}_{i}}$ using Equation (17) 25. end for 26. for each $i$ whale 27. ${r}_{1},{r}_{2}:$ Generate two random numbers between 0 and 1. 28. if ${\mathbf{r}}_{\mathbf{1}}<{\mathbf{r}}_{\mathbf{2}}$ 29. Update $\overrightarrow{{W}_{i}}\left(t+1\right)$ using Equation (19) 30. else 31. Update $\overrightarrow{{W}_{i}}\left(t+1\right)$ using Equation (20) 32. end if 33. if ${f}_{i}$<$of$ 34. $o{f}_{i}={f}_{i}$ 35. $\overrightarrow{O{W}_{i}}=\overrightarrow{{W}_{i}}$ 36. ${R}_{i}=0$ 37. else 38. $\overrightarrow{{W}_{i}}=\overrightarrow{O{W}_{i}}$ 39. ${R}_{i}++$ 40. end 41. end for 42. Update the best whale $\overrightarrow{{W}^{\ast}}$ with $\overrightarrow{W}\left(t+1\right)$ if better. 43. $t$++ end while. |
4. Experimental Settings
5. Results and Discussion
5.1. RTC France Cell
5.2. Photowatt-PWP201 Module
5.3. Kyocera KC200GT-204.6W
6. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter | LB | UB |
---|---|---|
${I}_{ph}\left(A\right)$ | $0.9{I}_{SC}$ | $1.1{I}_{SC}$ |
${I}_{sdi}\left(A\right),i\in 1:3$ | $1nA$ | $10\mu A$ |
${R}_{s}\left(\Omega \right)$ | $0$ | $0.5$ |
${R}_{sh}\left(\Omega \right)$ | $0$ | $500$ |
$a1$ | $1$ | $2$ |
$a2$ | $1.2$ | $2$ |
$a3$ | $1.4$ | $2$ |
Algorithms | ${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}1}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}2}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}3}\left(\mathit{A}\right)$ | ${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$ | ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$ | $\mathit{a}1$ | $\mathit{a}2$ | $\mathit{a}3$ | RMSE |
---|---|---|---|---|---|---|---|---|---|---|
AEO [23] | 0.75922 | 1.238 × 10^{−7} | 6.438 × 10^{−7} | 4.294 × 10^{−9} | 0.03505 | 85.16681 | 1.43068 | 1.68675 | 1.41757 | 0.00129701 |
ITLBO [20] | 0.76049 | 1.401 × 10^{−7} | 3.675 × 10^{−7} | 1.071 × 10^{−9} | 0.03676 | 57.43669 | 1.42406 | 1.66862 | 1.65840 | 0.00077984 |
ISA [25] | 0.76050 | 1.141 × 10^{−8} | 1.350 × 10^{−6} | 4.648 × 10^{−7} | 0.03899 | 70.30744 | 1.24034 | 1.78686 | 1.90577 | 0.00081148 |
HHO [10] | 0.75969 | 2.036 × 10^{−8} | 1.531 × 10^{−8} | 1.264 × 10^{−6} | 0.03720 | 108.84172 | 1.29243 | 1.51025 | 1.74264 | 0.00122689 |
WOA [18] | 0.76050 | 1.170 × 10^{−6} | 3.306 × 10^{−7} | 7.065 × 10^{−7} | 0.02743 | 429.58132 | 1.70700 | 1.70299 | 1.69958 | 0.00352126 |
RWOA | 0.76050 | 2.639 × 10^{−6} | 5.205 × 10^{−8} | 1.049 × 10^{−8} | 0.03841 | 64.45417 | 2.00000 | 1.34115 | 1.40000 | 0.00075626 |
HWOA | 0.76050 | 7.668 × 10^{−7} | 8.966 × 10^{−8} | 1.193 × 10^{−6} | 0.03795 | 60.85709 | 1.95480 | 1.37604 | 1.99836 | 0.00075148 |
Algorithms | AEO [23] | ITLBO [20] | ISA [25] | HHO [10] | WOA [18] | RWOA | HWOA |
---|---|---|---|---|---|---|---|
Best | 0.0012970061 | 0.0007798428 | 0.0008114755 | 0.0012268860 | 0.0035212594 | 0.0007562561 | 0.0007514822 |
Worst | 0.0058557573 | 0.0075862070 | 0.0049871863 | 0.0180751229 | 0.0182047656 | 0.0085551216 | 0.0058540672 |
Avg | 0.0038307293 | 0.0029920966 | 0.0027888508 | 0.0073241072 | 0.0098232146 | 0.0024644842 | 0.0010557685 |
SD | 0.0012625166 | 0.0014567314 | 0.0009987475 | 0.0037530593 | 0.0025490806 | 0.0016299982 | 0.0007712976 |
Time (s) | 1.8484064800 | 2.4445874020 | 1.1383867480 | 4.2349410020 | 2.2107720220 | 2.5196325380 | 1.3130094500 |
Rank | 5 | 4 | 3 | 6 | 7 | 2 | 1 |
Points | ITLBO [20] | HWOA | Points | ITLBO [20] | HWOA |
---|---|---|---|---|---|
1 | 0.00042140 | 0.00059436 | 14 | 0.00063736 | 0.00094555 |
2 | 0.00024578 | 0.00014751 | 15 | 0.00043540 | 0.00023563 |
3 | 0.00052240 | 0.00049222 | 16 | 0.00018622 | 0.00019598 |
4 | 0.00060074 | 0.00056986 | 17 | 0.00105762 | 0.00087708 |
5 | 0.00112908 | 0.00104595 | 18 | 0.00084037 | 0.00055163 |
6 | 0.00107938 | 0.00095619 | 19 | 0.00055482 | 0.00082578 |
7 | 0.00002304 | 0.00016786 | 20 | 0.00050983 | 0.00065411 |
8 | 0.00088337 | 0.00074426 | 21 | 0.00067192 | 0.00064497 |
9 | 0.00040942 | 0.00031250 | 22 | 0.00000501 | 0.00017199 |
10 | 0.00032050 | 0.00030689 | 23 | 0.00092424 | 0.00113327 |
11 | 0.00089763 | 0.00079535 | 24 | 0.00060347 | 0.00048777 |
12 | 0.00084004 | 0.00061514 | 25 | 0.00148841 | 0.00135363 |
13 | 0.00156924 | 0.00125975 | 26 | 0.00075850 | 0.00117864 |
RTC | PWP201 | KC200GT | ||||
---|---|---|---|---|---|---|
Algorithms | h | p-Value | h | p-Value | h | p-Value |
HWOA vs. AEO | 1 | 3.1949811 × 10^{−16} | 1 | 5.1397061 × 10^{−10} | 1 | 5.6387345 × 10^{−17} |
HWOA vs. ITLBO | 1 | 2.7981570 × 10^{−14} | 1 | 9.6808330 × 10^{−5} | 1 | 3.5812605 × 10^{−16} |
HWOA vs. ISA | 1 | 3.1940068 × 10^{−15} | 1 | 7.2511469 × 10^{−14} | 1 | 5.0379086 × 10^{−16} |
HWOA vs. HHO | 1 | 3.7391900 × 10^{−17} | 1 | 2.1975265 × 10^{−17} | 1 | 5.0154613 × 10^{−17} |
HWOA vs. WOA | 1 | 7.9688116 × 10^{−18} | 1 | 9.5403423 × 10^{−18} | 1 | 8.4619555 × 10^{−18} |
HWOA vs. RWOA | 1 | 2.1162998 × 10^{−10} | 1 | 1.2795224 × 10^{−6} | 1 | 3.9885266 × 10^{−6} |
Algorithms | ${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}1}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}2}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}3}\left(\mathit{A}\right)$ | ${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$ | ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$ | $\mathit{a}1$ | $\mathit{a}2$ | $\mathit{a}3$ | RMSE |
---|---|---|---|---|---|---|---|---|---|---|
AEO [23] | 1.03167 | 2.299 × 10^{−6} | 4.908 × 10^{−6} | 2.116 × 10^{−7} | 0.03423 | 23.20661 | 1.31099 | 1.98340 | 1.87961 | 0.00212391 |
ITLBO [20] | 1.03094 | 3.314 × 10^{−6} | 8.137 × 10^{−8} | 2.028 × 10^{−9} | 0.03361 | 26.15666 | 1.34608 | 1.80984 | 1.70548 | 0.00215662 |
ISA [25] | 1.03033 | 2.399 × 10^{−6} | 3.636 × 10^{−7} | 1.082 × 10^{−9} | 0.03436 | 26.49586 | 1.32168 | 1.37055 | 1.95677 | 0.00216172 |
HHO [10] | 1.02678 | 2.052 × 10^{−6} | 2.572 × 10^{−6} | 2.259 × 10^{−6} | 0.03163 | 78.98323 | 1.36586 | 1.44777 | 1.51062 | 0.00313249 |
WOA [18] | 1.02626 | 5.197 × 10^{−6} | 2.776 × 10^{−6} | 5.301 × 10^{−6} | 0.03173 | 474.24080 | 1.97039 | 1.94977 | 1.40202 | 0.00340767 |
RWOA | 1.03170 | 3.101 × 10^{−6} | 1.000 × 10^{−9} | 1.000 × 10^{−9} | 0.03376 | 23.59534 | 1.33896 | 2.00000 | 2.00000 | 0.00209360 |
HWOA | 1.03170 | 2.594 × 10^{−6} | 1.005 × 10^{−9} | 1.000 × 10^{−9} | 0.03436 | 22.16526 | 1.32049 | 2.00000 | 1.99996 | 0.00205067 |
Algorithms | AEO [23] | ITLBO [20] | ISA [25] | HHO [10] | WOA [18] | RWOA | HWOA |
---|---|---|---|---|---|---|---|
Best | 0.0021239070 | 0.0021566228 | 0.0021617217 | 0.0031324927 | 0.0034076744 | 0.0020936002 | 0.0020506744 |
Worst | 0.0045855675 | 0.0039748826 | 0.0049328947 | 0.0133624679 | 0.0339653392 | 0.0048751104 | 0.0037856010 |
Avg | 0.0031909775 | 0.0028101343 | 0.0034395527 | 0.0049863792 | 0.0092443412 | 0.0029661832 | 0.0024630570 |
SD | 0.0005529085 | 0.0004478418 | 0.0005988061 | 0.0021315308 | 0.0066477865 | 0.0005826925 | 0.0003765776 |
Time (s) | 2.0259489840 | 2.5425894060 | 1.0690436680 | 4.1146105580 | 2.1864761820 | 2.5288949540 | 1.1371688640 |
Rank | 4 | 2 | 5 | 6 | 7 | 3 | 1 |
Points | AEO [23] | HWOA | Points | AEO [23] | HWOA |
---|---|---|---|---|---|
1 | 0.00202612 | 0.00196520 | 14 | 0.00070850 | 0.00107830 |
2 | 0.00150929 | 0.00155811 | 15 | 0.00051425 | 0.00072990 |
3 | 0.00205488 | 0.00218585 | 16 | 0.00206436 | 0.00208065 |
4 | 0.00001616 | 0.00017725 | 17 | 0.00234223 | 0.00215810 |
5 | 0.00211195 | 0.00188234 | 18 | 0.00129039 | 0.00094117 |
6 | 0.00405118 | 0.00381960 | 19 | 0.00115746 | 0.00069973 |
7 | 0.00398503 | 0.00379324 | 20 | 0.00015724 | 0.00066030 |
8 | 0.00178349 | 0.00167704 | 21 | 0.00102009 | 0.00150851 |
9 | 0.00011291 | 0.00009277 | 22 | 0.00295107 | 0.00337354 |
10 | 0.00333524 | 0.00316378 | 23 | 0.00028751 | 0.00060339 |
11 | 0.00363641 | 0.00331901 | 24 | 0.00020045 | 0.00002180 |
12 | 0.00327006 | 0.00285082 | 25 | 0.00041889 | 0.00039876 |
13 | 0.00208071 | 0.00163878 | 26 | 0.00194215 | 0.00209381 |
Algorithms | ${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}1}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}2}\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{d}3}\left(\mathit{A}\right)$ | ${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$ | ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$ | $\mathit{a}1$ | $\mathit{a}2$ | $\mathit{a}3$ | RMSE |
---|---|---|---|---|---|---|---|---|---|---|
AEO [23] | 8.13273 | 1.457 × 10^{−7} | 6.419 × 10^{−7} | 5.324 × 10^{−8} | 0.00317 | 142.95143 | 1.33878 | 1.99886 | 1.67833 | 0.06162560 |
ITLBO [20] | 8.09277 | 8.870 × 10^{−8} | 2.782 × 10^{−9} | 5.431 × 10^{−7} | 0.00328 | 347.12120 | 1.30285 | 1.75460 | 1.88985 | 0.06196293 |
ISA [25] | 8.13193 | 1.794 × 10^{−8} | 1.432 × 10^{−6} | 4.577 × 10^{−9} | 0.00394 | 12.38224 | 1.20240 | 1.86530 | 1.53053 | 0.05929216 |
HHO [10] | 8.13923 | 1.000 × 10^{−9} | 1.956 × 10^{−8} | 2.460 × 10^{−8} | 0.00418 | 138.06531 | 1.06811 | 1.26960 | 1.46177 | 0.05470326 |
WOA [18] | 8.15932 | 3.135 × 10^{−9} | 1.579 × 10^{−9} | 4.621 × 10^{−7} | 0.00277 | 500.00000 | 2.00000 | 2.00000 | 1.42904 | 0.07293691 |
RWOA | 8.20030 | 1.000 × 10^{−9} | 1.310 × 10^{−9} | 1.053 × 10^{−9} | 0.00460 | 2.65765 | 1.04687 | 2.00000 | 1.94552 | 0.02821562 |
HWOA | 8.20186 | 1.000 × 10^{−9} | 1.000 × 10^{−9} | 1.000 × 10^{−9} | 0.00459 | 2.63009 | 1.04687 | 2.00000 | 1.69348 | 0.02822241 |
Algorithms | AEO [23] | ITLBO [20] | ISA [25] | HHO [10] | WOA [18] | RWOA | HWOA |
---|---|---|---|---|---|---|---|
Best | 6.16256 × 10^{−2} | 6.19629 × 10^{−2} | 5.92922 × 10^{−2} | 5.47033 × 10^{−2} | 7.29369 × 10^{−2} | 2.82156 × 10^{−2} | 2.82224 × 10^{−2} |
Worst | 1.20377 × 10^{−1} | 1.18683 × 10^{−1} | 1.11085 × 10^{−1} | 2.09556 × 10^{−1} | 2.53518 × 10^{−1} | 1.25166 × 10^{−1} | 8.74320 × 10^{−2} |
Avg | 9.26133 × 10^{−2} | 8.08383 × 10^{−2} | 8.24407 × 10^{−2} | 1.15432 × 10^{−1} | 1.60725 × 10^{−1} | 6.62490 × 10^{−2} | 4.86870 × 10^{−2} |
SD | 1.45518 × 10^{−2} | 1.12456 × 10^{−2} | 1.15311 × 10^{−2} | 3.13564 × 10^{−2} | 4.59415 × 10^{−2} | 1.98688 × 10^{−2} | 1.29875 × 10^{−2} |
Time (s) | 2.13778639 | 2.59146917 | 0.92115676 | 3.92217410 | 2.16986671 | 2.48034050 | 1.06056147 |
Rank | 5 | 3 | 4 | 6 | 7 | 2 | 1 |
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Abdel-Basset, M.; Mohamed, R.; El-Fergany, A.; Askar, S.S.; Abouhawwash, M. Efficient Ranking-Based Whale Optimizer for Parameter Extraction of Three-Diode Photovoltaic Model: Analysis and Validations. Energies 2021, 14, 3729. https://doi.org/10.3390/en14133729
Abdel-Basset M, Mohamed R, El-Fergany A, Askar SS, Abouhawwash M. Efficient Ranking-Based Whale Optimizer for Parameter Extraction of Three-Diode Photovoltaic Model: Analysis and Validations. Energies. 2021; 14(13):3729. https://doi.org/10.3390/en14133729
Chicago/Turabian StyleAbdel-Basset, Mohamed, Reda Mohamed, Attia El-Fergany, Sameh S. Askar, and Mohamed Abouhawwash. 2021. "Efficient Ranking-Based Whale Optimizer for Parameter Extraction of Three-Diode Photovoltaic Model: Analysis and Validations" Energies 14, no. 13: 3729. https://doi.org/10.3390/en14133729